标签: baseband

Fourier fits But it gets worse. In 1822, Joseph Fourier released his Théorie analytique de la chaleur ( The Analytic Theory of heat ). That seminal work has tormented generations of electrical engineering students (and no doubt others). Discontinuous functions—like square waves—are very resistant to mathematical analysis with calculus unless one does horrible things like use unit step functions. But Fourier showed that one can represent many of these periodic functions as the sum of sine waves of different amplitudes and frequencies. The Fourier Series for a square wave is:   The series goes on forever, so all square waves have frequency components going to infinity. However, the amplitude of these decrease rapidly due to the division by an ever-larger odd number. The point, though, is that the "frequency" of a square wave is composed of many frequencies higher than that of the baseband. Pulses, like the ones that race around every digital board, the ones we probe with our scopes and logic analyzers, are square-wave-ish. The good news is that they're not perfect square waves: obviously, with the exception of clocks, they rarely have a 50% duty cycle. Pulses are also, happily, imperfect. Fourier's analysis assumed that the signal transitions between zero and one instantaneously. In the real world every pulse has a finite rise and fall time. If T r is the rise time, then the frequency components above F in the following formula will be so far down they're not important:   This does mean that, assuming a 1-nsec rise time, even if your clock is ticking along at a leisurely rate about the same as a Florida old-timer's speedometer, the signals have significant frequencies up to 500MHz. Those unseen but very real frequency components will interact with the scope probe. Ad hoc formulas Long troubleshooting sessions often see a board covered with connections to test equipment, data loggers, etc. Long lengths of wire-wrap wire get soldered between a node and an instrument. These connections all change the AC properties of the nodes by adding inductance and capacitance. Here are some useful formulas with which one can estimate the effects. These are derived from book High-Speed Digital Design (Howard Johnson and Martin Graham, 1993 PTR Prentice-Hall Inc, Englewood Cliffs, NJ), and there's much more useful data in that book. Most of us use multilayer PCBs that have one or more ground and power planes. Solder a wire to a node and drape it across the board as it runs to a scope or other instrument, and you'll add capacitance. If d is the diameter of the wire in inches, h is the height above the PCB, and l is the length of the wire, then the capacitance in pF is:   (A better solution is to run the wire straight up from the node, perpendicular to the PCB.) AWG 30 wire-wrap wire is 0.0124 inches in diameter. Typical hook-up wires are AWG 20 (0.036 inches), AWG 22 (0.028 inches), and AWG 24 (0.022 inches). The inductance of the same wire in nanohenries is:   The inductance of a round loop of wire (for example a scope probe's ground lead) in nH is, if d is the diameter of the wire and x is the diameter of the loop:   Next month we'll look at some real-world data.